The point of this lab is to see the relationship between the driving frequency and the reactive capacitance. When we increase the frequency, the total circuit impedance will decrease, making the the amplitude increase. We also find out that the voltage across capacitor and voltage across resistor is the same when their resistance are the same. To better understand this concept, we had to use the oscilloscope and adjust the driving frequency until both capacitor and resistor voltage amplitude are the same. After doing the experiment, we found the slope to be 8.86E-8 which equals to be C to find the capacitance.
Procedure
We hooked the system with a 10000 ohms resistor and tried to find the value of our capacitance using the equation XC=1/(WdriveC). We first had to rearrange the equation so it can match with the values being shown on the oscillator, which is the (1/R)(Vresistor amplitude/capacitor amplitude)=C*WD where you look at the graph like y=mx. Using that equation, we knew we had to collect data for the voltages of both resistor and capacitor and the driving frequency. By changing the driving frequency, the voltages amplitude changes. And because it was a sinusoidal graph, we had to multiply the driving frequency by 2pi. After getting five data points, we graphed it to match (1/R)(Vresistor amplitude/capacitor amplitude vs WD. By doing this, we can see that the capacitor value is the slope of the graph.
We also had make sure all the values were shown correctly on the oscillator by pressing the different types of mode. All systems were set up in a series. And the oscilloscope was set up parallel to find the voltage across the system.
Results: Resistor= 10,000 ohms
Because the equation called for it, we had to calculate (1/R)(Vresistor amplitude/capacitor amplitude). For example, we had to put the values in like this: (1/10000)(1.5/2.5). That would be on our y-axis, with the Wd*2pi on our X axis. After graphing it out, we found the slope to be 8.86E-8 Farads by graphing by hand; we checked the values by plugging the values into the calculator and finding slope. The intersection B value was 9.189E-6
Discussion
The value that we found is 8.86E-8 farads for the capacitance of the system. However if we think about the numbers individually, with each changing frequency, the C value is also changing. Technically we are measuring the rate at which the capacitance is changing or as a whole. Comparing the slope to the individual calculated C, there was a big difference. There might be a source of error along the way from the connection and how we gathered our data. Because the graph was slightly moving, we could only get a point that averaged and popped up most often.
Open-Ended Discussion
From Week 8, we had to find the equation for capacitance of a hand-made capacitor, using the given equation of C=KOA/D, where &o is 8.85E-12 C2/(N*m2). We also needed to find the K value, which is the dielectric constant. To do this, we had to find the capacitance first, by using the (1/R)(Vresistor amplitude/capacitor amplitude)=C*WD. We just needed to change out the capacitor within the series with the cardboard capacitor, which was made of two aluminum board with a cardboard in the center. The diameter between the two aluminum pieces were 1.4mm and the boards were 41 x 41 cm, which was later changed to the scientific units. Then using the oscilloscope, we just needed to set up so we can change the frequency and be able to see the Voltage amplitudes for resistance and capacitor. The resistor in the system was 10000 ohms.
This means we had to change the driving frequency and then use the voltage amplitude given. We then graphed it out on the calculator to find the slope of the graph, which turned out to be 1.462E-9 nano farads. Knowing the capacitance, we just needed to know the area of the boards, which was 1681cm. With all the values calculated and given, we just had to plug it into the equation to find the K value, which came out to be 1.376, with no units.
We knew we had to use that method because we didn’t have other materials to just find the capacitance and the resistance, and because it was changing when we changed the driving frequency, we knew that we needed more data to find the capacitance.
